Ends and Tangles

Abstract

We show that an arbitrary infinite graph can be compactified by its ${\aleph }_0$-tangles in much the same way as the ends of a locally finite graph compactify it in its Freudenthal compactification. In general, the ends then appear as a subset of its ${\aleph }_0$-tangles. The ${\aleph }_0$-tangles of a graph are shown to form an inverse limit of the ultrafilters on the sets of components obtained by deleting a finite set of vertices. The ${\aleph }_0$-tangles that are ends are precisely the limits of principal ultrafilters. The ${\aleph }_0$-tangles that correspond to a highly connected part, or ${\aleph }_0$-block, of the graph are shown to be precisely those that are closed in the topological space of its finite-order separations.

References updated, no final

References in corpus (1)

• Excluding a full grid minor

Cited by in corpus (4)

• Tangle-tree duality in abstract separation systems
• Duality theorems for stars and combs I: Arbitrary stars and combs
• Trees of tangles in infinite separation systems
• Efficiently distinguishing all tangles in locally finite graphs